The Fallwing family loves gummy bears and pickled strawberries. Their monthly GB/PS utility function is given by U(x, y) = 13lnx + 7lny, where x = the number of pounds of gummy bears and y = the number of pounds of pickled strawberries that they consume each month. (a) Find the quantities of gummy bears and pickled strawberries that the Fallwin family should consume to maximize their GB/PS utility, given that gummy bears cost $8.00/pound, pickled strawberries cost $6.00/pound, and the Fallwing's monthly GB/PS budget is $480.00. What is their maximum monthly GB/PS utility? (b) Show how you know that the GB/PS utility that you found in (a) is the Fallwing's absolute maximum.
The Fallwing family loves gummy bears and pickled strawberries. Their monthly GB/PS utility function is given by U(x, y) = 13lnx + 7lny, where x = the number of pounds of gummy bears and y = the number of pounds of pickled strawberries that they consume each month. (a) Find the quantities of gummy bears and pickled strawberries that the Fallwin family should consume to maximize their GB/PS utility, given that gummy bears cost $8.00/pound, pickled strawberries cost $6.00/pound, and the Fallwing's monthly GB/PS budget is $480.00. What is their maximum monthly GB/PS utility? (b) Show how you know that the GB/PS utility that you found in (a) is the Fallwing's absolute maximum.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![## The Fallwing Family's Consumption of Gummy Bears and Pickled Strawberries
### Utility Maximization Problem
The Fallwing family enjoys consuming gummy bears and pickled strawberries. Their monthly utility from these goods can be represented by the utility function:
\[ U(x, y) = 13 \ln x + 7 \ln y \]
where:
- \( x \) represents the number of pounds of gummy bears consumed per month, and
- \( y \) represents the number of pounds of pickled strawberries consumed per month.
### Problem Statement:
#### (a) Find the Optimal Consumption Quantities:
Determine the quantities of gummy bears and pickled strawberries that the Fallwing family should consume to maximize their utility, given the following information:
- The cost of gummy bears is $8.00 per pound.
- The cost of pickled strawberries is $6.00 per pound.
- The Fallwing family's monthly budget for these items is $480.00.
What is the maximum monthly utility that the Fallwing family can achieve?
#### (b) Verify the Maximum Utility:
Show how it can be verified that the GB/PS utility obtained in part (a) is indeed the Fallwing family's absolute maximum.
### Solution Approach:
To solve this problem:
1. **Formulate the Budget Constraint:**
\[ 8x + 6y = 480 \]
2. **Use the Utility Function:**
\[ U(x, y) = 13 \ln x + 7 \ln y \]
3. **Apply the Method of Lagrange Multipliers:**
First, set up the Lagrangian:
\[ \mathcal{L}(x, y, \lambda) = 13 \ln x + 7 \ln y + \lambda (480 - 8x - 6y) \]
4. **Take Partial Derivatives and Set to Zero:**
\[ \frac{\partial \mathcal{L}}{\partial x} = \frac{13}{x} - 8 \lambda = 0 \]
\[ \frac{\partial \mathcal{L}}{\partial y} = \frac{7}{y} - 6 \lambda = 0 \]
\[ \frac{\partial \mathcal{L}}{\partial \lambda} = 480 - 8x - 6y = 0 \]
5. **Solve the System of Equations to Find \( x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c81efc3-2b49-4b96-98ce-19350f40aab1%2F52e3a037-ff3b-4edf-8644-b036cb4c508f%2Fqludqk_processed.png&w=3840&q=75)
Transcribed Image Text:## The Fallwing Family's Consumption of Gummy Bears and Pickled Strawberries
### Utility Maximization Problem
The Fallwing family enjoys consuming gummy bears and pickled strawberries. Their monthly utility from these goods can be represented by the utility function:
\[ U(x, y) = 13 \ln x + 7 \ln y \]
where:
- \( x \) represents the number of pounds of gummy bears consumed per month, and
- \( y \) represents the number of pounds of pickled strawberries consumed per month.
### Problem Statement:
#### (a) Find the Optimal Consumption Quantities:
Determine the quantities of gummy bears and pickled strawberries that the Fallwing family should consume to maximize their utility, given the following information:
- The cost of gummy bears is $8.00 per pound.
- The cost of pickled strawberries is $6.00 per pound.
- The Fallwing family's monthly budget for these items is $480.00.
What is the maximum monthly utility that the Fallwing family can achieve?
#### (b) Verify the Maximum Utility:
Show how it can be verified that the GB/PS utility obtained in part (a) is indeed the Fallwing family's absolute maximum.
### Solution Approach:
To solve this problem:
1. **Formulate the Budget Constraint:**
\[ 8x + 6y = 480 \]
2. **Use the Utility Function:**
\[ U(x, y) = 13 \ln x + 7 \ln y \]
3. **Apply the Method of Lagrange Multipliers:**
First, set up the Lagrangian:
\[ \mathcal{L}(x, y, \lambda) = 13 \ln x + 7 \ln y + \lambda (480 - 8x - 6y) \]
4. **Take Partial Derivatives and Set to Zero:**
\[ \frac{\partial \mathcal{L}}{\partial x} = \frac{13}{x} - 8 \lambda = 0 \]
\[ \frac{\partial \mathcal{L}}{\partial y} = \frac{7}{y} - 6 \lambda = 0 \]
\[ \frac{\partial \mathcal{L}}{\partial \lambda} = 480 - 8x - 6y = 0 \]
5. **Solve the System of Equations to Find \( x
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